1. Multiple-Angle and Product-to-Sum Formulas
Skill 34b

2. Objectives…
Use multiple–angle formulas to rewrite and evaluate trigonometric functions
Use power–reducing formulas to rewrite and evaluate trigonometric functions
Use half–angle formulas to rewrite and evaluate trigonometric functions
Use product–to–sum and sum–to–product formulas to rewrite and evaluate trigonometric functions.

3. Categories of Multiple Angles
Functions of multiple angles
Squares of trigonometric functions
Functions of half–angles
Products of trigonometric functions

5. Example–Solving a Multiple –Angle Equation
Solve 2 cos x + sin 2x = 0.
Solution:
2cos x + sin 2x = 0
2 cos x + 2 sin x cos x = 0
2 cosx(1 + sin x) = 0
2 cos x = sin x = 0
cos x = sin x = –1
Write original equation.
Double–angle formula
Factor.
Set factors equal to zero
Isolate trig. functions.

6. Example–Solution x = + 2n and x = + 2n where n is an integer.
Solutions in [0, 2)
General solution

8. Example–Reducing a Power
Rewrite sin4x as a sum of first powers of the cosines of multiple angles.
Solution:
sin4x = (sin2x)2 =
= ¼ (1 – 2 cos2x + cos22x)

9. Example–Solution

11. Example–Using a Half–Angle Formula
Find the exact value of sin 105.
Solution: 105 = ½ * 210
105 lies in Quadrant II, you have
sin  is positive in Quadrant II.

12. Example–Solving a Trigonometric Equation
Find all solutions of in the interval [0, 2).
Solution:
1 + cos2x = 1 + cos x
cos2 x – cos x = 0

13. Example–Solution cos x(cos x – 1) = 0
Set the factors cos x and cos x – 1 equal to zero
The solutions in the interval [0,2 ) are
x = , x = , and x = 0.
Factor.

16. Example–Writing Products as Sums
Rewrite the product as a sum or difference. cos 5x sin 4x
Solution: Using the appropriate product-to-sum formula,
cos 5x sin 4x = ½[sin(5x + 4x) – sin(5x – 4x)]
= ½sin 9x – ½sin x.

17. Example–Using a Sum–to–Product Formula
Find the exact value of cos 195° + cos 125°.
Solution: Using the appropriate sum-to-product formula, you obtain
cos 195° + cos 105° =
= 2 cos 150° cos 45°

18. 34b Multiple-Angle & Product-to-Sum Formulas
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